\section{Diameter-based objective}
\label{sec:theory2}
In this section, we mention theoretical results for {\it Diameter-sTF} and {\it Diameter-mTF}. We show that these problems are NP-hard (note that this does not follow from any previous work); the proof is presented in Appendix~\ref{sec:minDiaNPProof}.
%Note that the NP-hardness of {\it Diameter-sTF} does not follow from any previous work. However, due to space constraints the proofs are omitted in this draft.
\begin{theorem} 
\label{Thm:minDiaNP}
{\it Diameter-sTF} and {\it Diameter-mTF} are NP-complete.
\end{theorem}
%Details of the proof are mentioned in the Appendix ~\ref{sec:minDiaNPProof}.


%The algorithm {\it MinDiameter} generalizes the algorithm {\it RarestFirst} described in ~\cite{LLT}. 
We further present an algorithm {\it MinDiameter} (Algorithm ~\ref{algo:minDia})  which is an extension of {\it RarestFirst} in~\cite{LLT}, and prove that it achieves a 2-approximation factor (proof in Appendix~\ref{sec:minDia2Approx}). The pseudocode is presented in Appendix~\ref{sec:minDiaAlgo} but we describe the algorithm here. For each individual, say $i_r \in S(a_{rare})$ where $a_{rare}$ is the rarest skill (the skill with the minimum size support set $S$), and for each skill $a_i \in {\cal T}$, the algorithm finds the distance to all the nodes in the support set $S(a_i)$. Then, for each support set $S(a_i)$, it chooses the $k_i$-size subset of $S(a_i)$ such that the maximum shortest path distance between $i_r$ and the nodes in this subset is minimum among all $k_i$-size subsets of $S(a_i)$. We call this distance as $k_i$-th shortest distance between $i_r$ and $S(a_i)$ and denote it as $d_k(i_r, S(a_i), k_i)$. Further, we denote the set of $k_i$ shortest paths between $i_r$ and each of the nodes belonging to the corresponding $k_i$-size subset of $S(a_i)$ as $Path_k(i_r, S(a_i), k_i)$. Thus, for each $i_r \in S(a_{rare})$ the algorithm has identified $k_i$ nodes of skill $a_i$, thereby forming a possible solution team that satisfies the constraints. Finally, the algorithm then picks one of these solutions that has minimum diameter. 
%The algorithm is presented below in detail. 
The time complexity of the algorithm {\it MinDiameter}, assuming that all pairs shortest paths are pre-computed, is $O(n^2)$.

\begin{theorem} \label{thm:diamapprox}
For any graph distance function with triangle inequality, the algorithm {\it MinDiameter} achieves an approximation factor of $2$ for the {\it Diameter-sTF} and {\it Diameter-mTF} problems.
\end{theorem}
%Details of the proof are mentioned in the Appendix ~\ref{sec:minDia2Approx}.